{
  "id": "math/0606625",
  "version": "v1",
  "published": "2006-06-24T13:57:00.000Z",
  "updated": "2006-06-24T13:57:00.000Z",
  "title": "A Central Limit Theorem for biased random walks on Galton-Watson trees",
  "authors": [
    "Yuval Peres",
    "Ofer Zeitouni"
  ],
  "comment": "34 pages, 4 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let ${\\cal T}$ be a rooted Galton-Watson tree with offspring distribution $\\{p_k\\}$ that has $p_0=0$, mean $m=\\sum kp_k>1$ and exponential tails. Consider the $\\lambda$-biased random walk $\\{X_n\\}_{n\\geq 0}$ on ${\\cal T}$; this is the nearest neighbor random walk which, when at a vertex $v$ with $d_v$ offspring, moves closer to the root with probability $\\lambda/(\\lambda+d_v)$, and moves to each of the offspring with probability $1/(\\lambda+d_v)$. It is known that this walk has an a.s. constant speed $\\v=\\lim_n |X_n|/n$ (where $|X_n|$ is the distance of $X_n$ from the root), with $\\v>0$ for $ 0<\\lambda<m$ and $\\v=0$ for $\\lambda \\ge m$. For all $\\lambda \\le m$, we prove a quenched CLT for $|X_n|-n\\v$. (For $\\lambda>m$ the walk is positive recurrent, and there is no CLT.) The most interesting case by far is $\\lambda=m$, where the CLT has the following form: for almost every ${\\cal T}$, the ratio $|X_{[nt]}|/\\sqrt{n}$ converges in law as $n \\to \\infty$ to a deterministic multiple of the absolute value of a Brownian motion. Our approach to this case is based on an explicit description of an invariant measure for the walk from the point of view of the particle (previously, such a measure was explicitly known only for $\\lambda=1$) and the construction of appropriate harmonic coordinates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-06-24T13:57:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "60F05",
      "60J80",
      "82C41"
    ],
    "keywords": [
      "biased random walk",
      "central limit theorem",
      "nearest neighbor random walk",
      "appropriate harmonic coordinates",
      "moves closer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......6625P"
    }
  }
}